Gauge Theory and Geometry in Dimensions Three and Four

这个项目的目的是进一步发展和应用从规范理论中产生的各种结和三流形的Floer同调理论。特别是，首席研究员将进一步发展一个Floer同源理论的结，从瞬子与余维2奇点，并将调查其关系到Khovanov同源。一个光谱序列将建立有关的两个同源性理论。 他将调查是否可以证明非平凡的弗洛尔理论所产生的奇异瞬子，以期表明霍瓦诺夫同源检测unknot。 缝合流形的瞬子同调理论，最近开发的PI和Mrowka，将进一步探讨。将证明由此产生的链接的Floer同调恢复多变量亚历山大多项式。在类似的精神下，封闭三维流形的瞬子同调将被证明恢复雷德迈斯特挠率。 PI将定义transverselink的不变量：在三球或其他3流形中横向于标准接触结构的链接。这些不变量将驻留在链接弗洛尔同调群产生的瞬子理论缝合流形。拓扑学是定性研究空间及其连通性。 它的重要性在上个世纪之交被法国数学家庞加莱认识到，在他对支配三体系统运动的运动定律的研究中，如地球，月球和太阳根据牛顿定律运动。 在过去的20年里，拓扑学已经在蛋白质和DNA的打结等问题以及高能物理的现代理论中得到了应用。 三维空间的拓扑结构，与高维空间的拓扑结构相反，特别微妙。 近年来，大量的新技术已经被引入到研究三维空间中出现的现象。这个项目将进一步发展这些新技术，并将探索它们之间的神秘关系。通过这样做，该项目将加深我们对拓扑学及其与数学和科学其他领域的相互作用的理解。同时，该项目将培训研究生，并向该领域的研究人员传播成果。

The aim of this project is to further develop and apply the various Floer homology theories of knots and three-manifolds that arise from gauge theories. In particular, the Principal Investigator will further develop a Floer homology theory for knots, arising from instantons with codimension-2 singularities, and will investigate its relationship to Khovanov homology. A spectral sequence will be established relating the two homology theories. He will investigate whether one can prove the non-triviality of the Floer theory arising from singular instantons, with a view towards showing that Khovanov homology detects the unknot. The instanton homology theory of sutured manifolds, recently developed by the PI and Mrowka, will be further explored. It will be proved that the resulting Floer homology of links recovers the multi-variable Alexander polynomial. In a similar spirit, the instanton homology of closed 3-manifolds will be shown to recover the Reidemeister torsion. The PI will define invariants of transverselinks: links transverse to the standard contact structure in the three-sphere or other 3-manifolds. These invariants will reside in the link Floer homology groups arising from the instanton theory on sutured manifolds.Topology is the qualitative study of space and its connectedness. Its importance was recognized at the turn of the last century by the French mathematician Poincare, during his investigation of the laws of motion that govern the movement of a three-body system such as the Earth, Moon and Sun moving according to Newton's laws. In the past twenty years, topology has seen applications in questions such as the knotting of proteins and DNA, and in modern theories of high-energy physics. The topology of three-dimensional spaces, as opposed to those of higher dimension, is of particular subtlety. In recent years, a wealth of new techniques have been introduced to study the phenomena that arise in dimension three. This project will further develop these new techniques, and will explore the mysterious relationships between them. In so doing, the project will deepen our understanding of topology and its interaction with other areas of mathematics and science. At the same time, the project will train graduate students and disseminate results to researchers in the area.